Problem: What is the remainder when the sum $1 + 7 + 13 + 19 + \cdots + 253 + 259$ is divided by $6$?
Answer: First of all, we see that every term has a remainder of $1$ when divided by $6.$ Now, we just need to find how many terms there are. The nth term can be given by the expression $6n - 5.$ Therefore, we set $259 = 6n - 5$ to find $n = 44,$ thus there are $44$ terms in our sum. Thus, the remainder of the sum is the same as the remainder of $44$ when divided by $6,$ which is $\boxed{2}.$